My laptop randomly turns off

I've always been running on A/C power but lately my computer loses power for no reason. It's not that it shuts down, but instead it just turns off like it suddenly lost power. I thought maybe my charger was dying so I bought one from HP but that hasn't changed anything. My battery hasn't worked in a really, really long time so I never put it in the laptop anymore. I tried the computer in the outlets in my bedroom and in the library and I got the same results. Sometimes the power shuts off as the computer is booting, other times it lets me get to the home screen and I can play on the computer on a while before it loses power. I always keep my laptop on top of a chill pad. Although I've noticed my fan doesn't turn on when I turn on my computer. Is it suppose to? Also, I think I might have heard a noise from my fan. How can I be sure it's the fan?

I have a HP G60 120US. I bought it in 2008. It's running off Windows Vista Service Pack 2.
 
The laptop is definitely old enough for some service. It is possible that the cooling fan is worn out and/or dirty... if the fan is not running properly, the laptop will sense a over-heating condition and it will turn off. Have the laptop serviced and new batteries are cheap enough on EBay
 
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